I. INTRODUCTION
Micro-Doppler Effect in Radar:
Phenomenon, Model, and
Simulation Study
VICTOR C. CHEN
Naval Research Laboratory
FAYIN LI
SHEN-SHYANG HO
HARRY WECHSLER, Fellow, IEEE
George Mason University
When, in addition to the constant Doppler frequency shift
induced by the bulk motion of a radar target, the target or any
structure on the target undergoes micro-motion dynamics, such
as mechanical vibrations or rotations, the micro-motion dynamics
induce Doppler modulations on the returned signal, referred to
as the micro-Doppler effect. We introduce the micro-Doppler
phenomenon in radar, develop a model of Doppler modulations,
derive formulas of micro-Doppler induced by targets with
vibration, rotation, tumbling and coning motions, and verify
them by simulation studies, analyze time-varying micro-Doppler
features using high-resolution time-frequency transforms, and
demonstrate the micro-Doppler effect observed in real radar data.
Manuscript received March 1, 2003; revised July 1, 2004 and
March 3, 2005; released for publication August 5, 2005.
IEEE Log No. T-AES/42/1/870577.
Refereeing of this contribution was handled by L. M. Kaplan.
This work was supported in part by the Office of Naval Research
and the Missile Defense Agency.
Authors’ addresses: V. C. Chen, Radar Division, Naval Research
Laboratory, Code 5311, 4555 Overlook Ave. SW, Washington, D.C.
20375; F. Li, S-S. Ho, and H. Wechsler, Dept. of Computer Science,
George Mason University, Fairfax, VA 22030.
c 2006 IEEE
0018-9251/06/$17.00 °
2
When a radar transmits an electromagnetic signal
to a target, the signal interacts with the target and
returns back to the radar. Changes in the properties of
the returned signal reflect the characteristics of interest
for the target. When the target moves with a constant
velocity, the carrier frequency of the returned signal
will be shifted. This is known as the Doppler effect
[1]. For a mono-static radar where the transmitter and
the receiver are at the same location, the roundtrip
distance traveled by the electromagnetic wave is
twice the distance between the transmitter and the
target. The Doppler frequency shift is determined
by the wavelength of the electromagnetic wave and
the relative velocity between the radar and the target:
fD = ¡2¸V, where ¸ = c=f is the wavelength and V
is the relative velocity. If the radar is stationary, the
relative velocity V will be the velocity of the target
along the line of sight (LOS) of the radar, known as
the radial velocity. When the target is moving away
from the radar, the velocity is defined to be positive,
and as a consequence the Doppler shift is negative.
If the target or any structure on the target has
mechanical vibration or rotation in addition to
its bulk translation, it might induce a frequency
modulation on the returned signal that generates
sidebands about the target’s Doppler frequency
shift. This is called the micro-Doppler effect [2—4].
Radar signals returned from a target that incorporates
vibrating or rotating structures, such as propellers
of a fixed-wing aircraft, rotors of a helicopter, or
the engine compressor and blade assemblies of a jet
aircraft, contain micro-Doppler characteristics related
to these structures. The micro-Doppler effect enables
us to determine the dynamic properties of the target
and it offers a new approach for the analysis of target
signatures. Micro-Doppler features serve as additional
target features that are complementary to those made
available by existing methods. The micro-Doppler
effect can be used to identify specific types of
vehicles, and determine their movement and the speed
of their engines. Vibrations generated by a vehicle
engine can be detected by radar signals returned
from the surface of the vehicle. From micro-Doppler
modulations in the engine vibration signal, one can
distinguish whether it is a gas turbine engine of a tank
or the diesel engine of a bus.
The micro-Doppler effect was originally
introduced in coherent laser systems [3]. A coherent
laser radar system transmits electromagnetic waves at
optical frequencies and receives the backscattered light
waves from targets. A coherent system preserves the
phase information of the scattered waves with respect
to a reference wave and has greater sensitivity to any
phase variation. Because a half-wavelength change in
range can cause a 360± phase change, for a coherent
laser system with a wavelength of ¸ = 2 ¹m, 1 ¹m
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 1
JANUARY 2006
variation in range would cause a 360± phase
change.
In many cases, a target or a structure on the
target may have micro-motions, such as vibrations or
rotations. The source of rotations or vibrations might
be a rotating rotor of a helicopter, a rotating antenna
on a ship, mechanical oscillations in a bridge or a
building, an engine-induced vibrating surface, or other
causes. Micro-motion dynamics produce frequency
modulations on the back-scattered signal and would
induce additional Doppler changes to the constant
Doppler frequency shift of the bulk translational
motion. For a target that has only translation with a
constant velocity, the Doppler frequency shift induced
by translation is a time-invariant function. If the
target also undergoes a vibration or rotation, then the
Doppler frequency shift generated by the vibration
or rotation is a time-varying frequency function and
imposes a periodic time-varying modulation onto the
carrier frequency. Micro-motions yield new features
in the target’s signature that are distinct from its
signature in the absence of micro-motions.
For a pure periodic vibration or rotation,
micro-motion dynamics generate sideband Doppler
frequency shifts about the Doppler shifted central
carrier frequency. The modulation contains harmonic
frequencies that depend on the carrier frequency,
the vibration or rotation rate, and the angle between
the direction of vibration, and the direction of the
incident wave. Because the frequency modulation is
a phase change in the signal, in order to extract useful
information from the modulation, coherent processing
must be used to carefully track the phase change.
For a vibration scatterer, if the vibration rate in
angular frequency is !v and the maximal displacement
of the vibration is Dv , the maximum Doppler
frequency variation is determined by maxffD g =
(2=¸)Dv !v . As a consequence, for very short
wavelengths, even with very low vibration rate !v any
micro vibration of Dv can cause large phase changes.
As a consequence the micro-Doppler frequency
modulation or the phase change with time can be
easily detected. A coherent laser radar operating at
1:5 ¹m wavelength, can achieve a velocity precision
¢V better than 1 mm/s, or a Doppler resolution of
¢fD = 2¢V=¸ = 1:33 KHz.
Because the micro-Doppler effect is sensitive
to the operating frequency band, for radar systems
operating at microwave frequency bands, the
phenomenon may also be observable if the product
of the target’s vibration rate and the displacement of
the vibration is high enough. For a radar operating at
X-band with a wavelength of 3 cm, a vibration rate
of 15 Hz with a displacement of 0.3 cm can induce a
detectable maximum micro-Doppler frequency shift
of 18.8 Hz. If the radar is operated at L-band with a
wavelength of 10 cm, to achieve the same maximum
micro-Doppler shift of 18.8 Hz at the same vibration
rate of 15 Hz, the required displacement must be
1 cm, which may be too large in practice. Therefore,
at lower radar frequency bands, the detection of the
micro-Doppler modulation generated by vibration
may not be possible. The micro-Doppler generated
by rotations, such as rotating rotor blades, however,
may be detectable because of their longer rotating
arms and, thus, higher tip speeds. For example,
UHF-band (300—1,000 MHz) radar with a wavelength
of 0.6 m, when a helicopter’s rotor blade rotates
with a tip speed of 200 m/s, can induce a maximum
micro-Doppler frequency shift of 666 Hz that is
certainly detectable.
To analyze time-varying micro-Doppler frequency
features, the Fourier transform, which is unable to
provide time-dependent frequency information, is not
suitable. An efficient method to analyze time-varying
frequency features is to apply a high-resolution
time-frequency transform.
The contribution of this paper is that 1) a
model of the micro-Doppler effect is developed,
2) mathematical formulas of micro-Doppler
modulations induced by several typical basic
micro-motions are derived and verified by simulation
studies, 3) instead of using the conventional Fourier
transform, the high-resolution time-frequency
transform is used to analyze time-varying
micro-Doppler features, and 4) micro-Doppler effect
in radar is demonstrated using real radar data. In
Section II, we develop a model for analyzing the
micro-Doppler effect. In Section III, we briefly
introduce high-resolution time-frequency transforms
for analyzing time-varying frequency spectrum.
In Section IV, we apply the model for analyzing
micro-Doppler effect to several typical micro-motions
(vibration, rotation, tumbling, and coning) and verify
them using simulation studies. In Section V, we
demonstrate two examples of micro-Doppler effect
in radar observed in real radar data.
II.
MICRO-DOPPLER EFFECT INDUCED BY
MICRO-MOTION DYNAMICS
The micro-Doppler effect induced by
micro-motions of a target or structures on the target
can be derived from the theory of electromagnetic
back-scattering field. It can be mathematically
formulated by augmenting the conventional Doppler
effect analysis using micro-motions.
The characteristics of the electromagnetic
back-scattering field from a moving or an oscillating
target have been studied in both theory and
experiment [5—14]. Theoretical analysis indicates
that the translation of a target modulates the phase
function of the scattered electromagnetic waves. When
the target oscillates linearly and periodically, the
modulation generates sideband frequencies about the
frequency of the incident wave. A far electric field of
CHEN ET AL.: MICRO-DOPPLER EFFECT IN RADAR: PHENOMENON, MODEL, AND SIMULATION STUDY
3
Fig. 1. Geometry of translation target in far EM field.
a translated target can be derived as [8]
0
~ T (~r ) = expfjk~r0 ¢ (~uk ¡ ~ur )gE
~ (~r)
E
(1)
where k = 2¼=¸ is the wave number, ~uk is the
unit vector of the incidence wave, ~ur is the unit
~ (~r) is the
vector of the direction of observation, E
far electric field of the target before moving, ~r =
(U0 , V0 , W0 ) is the initial coordinates of the target in
the radar coordinates (U, V, W), ~r0 = (U1 , V1 , W1 ) is the
coordinates of the target after translation, and ~r =
~r0 +~r0 , where ~r0 is the translation vector, as illustrated
in Fig. 1.
From (1) we can see that the only difference in
the electric field before and after the translation is the
phase factor expfjk~r0 ¢ (~uk ¡ ~ur )g. If the translation is
a function of time ~r0 = ~r0 (t) = r0 (t)~uT , where ~uT is the
unit vector of the translation, the phase factor then
becomes
expfj©(t)g = expfjkr0 (t)~uT ¢ (~uk ¡ ~ur )g:
(2)
For back-scattering, the direction of observation is
opposite to the direction of the incidence wave, or
~uk = ¡~ur and thus
expfj©(t)g = expfj2kr0 (t)~uT ¢ ~uk g:
(3)
If the translation direction is perpendicular to the
direction of the incidence wave, the phase function
is zero and expf©(t)g = 1.
In general, when the radar transmits an
electromagnetic wave at a carrier frequency of f, the
radar received signal can be expressed as
~ (~r)j (4)
s(t) = expfj2kr0 (t)~uT ¢ ~uk g expf¡j2¼ftgjE
where the phase factor, expfj2kr0 (t)~uT ¢ ~uk g, defines
the modulation of the micro-Doppler effect caused
by the motion ~r0 (t). If the motion is a vibration
given by r0 (t) = A cos −t, the phase factor becomes a
periodic function of the time with an angular vibrating
frequency −
expfj©(t)g = expfj2kA cos −t~uT ¢ ~uk g:
(5)
The phase function can be mathematically
formulated by introducing micro-motions to augment
the conventional Doppler analysis. Let us represent
a target as a set of point scatterers that represent
the primary scattering centers on the target. The
point scattering model simplifies the analysis while
4
Fig. 2. Geometry of radar and target with translation and
rotation.
preserving the micro-Doppler features. For simplicity,
all scatterers are assumed to be perfect reflectors that
reflect all the energy intercepted.
As shown in Fig. 2, the radar is stationary and
located at the origin Q of the radar coordinate system
(U, V, W). The target is described in a local coordinate
system (x, y, z) attached to it and has translations and
rotations with respect to the radar coordinates. To
observe the target’s rotations, a reference coordinate
system (X, Y, Z) is introduced, which shares the same
origin with the target local coordinates and, thus, has
the same translation as the target but no rotation with
respect to the radar coordinates. The origin O of the
reference coordinates is assumed to be at a distance R0
from the radar.
Suppose the target is a rigid body that has
~ with respect to the radar and
translation velocity V
a rotation angular velocity ~!, which can be either
represented in the target local coordinate system
as !
~ = (!x , !y , !z )T , or represented in the reference
coordinate system as !
~ = (!X , !Y , !Z )T . Because the
motion of a rigid body can be represented by the
position of the body at two different instants of time,
a particle P of the body at instant of time t = 0 will
move to P 0 at instant of time t. The movement consists
of two steps: 1) translation from P to P 00 , as shown
¡¡!
~ , i.e., OO0 = V
~ t, and
in Fig. 2, with a velocity V
2) rotation from P 00 to P 0 with an angular velocity
!
~ . If we observe the movement in the reference
coordinate system, the particle P is located at
~r0 = (X0 , Y0 , Z0 )T , and the rotation from P 00 to P 0 is
described by a rotation matrix <t . Then, at time t the
location of P 0 will be at
¡¡!
¡¡!
(6)
~r = O0 P 0 = <t O0 P 00 = <t~r0 :
The range vector from the radar at Q to the particle at
P 0 can be derived as
¡¡!0 ¡! ¡¡!0 ¡¡0!0
~0 +V
~ t + <t~r0
QP = QO + OO + O P = R
(7)
and the scalar range becomes
~ t + <t~r0 k
~0 + V
r(t) = kR
(8)
where k ¢ k represents the Euclidean norm.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 1
JANUARY 2006
If the radar transmits a sinusoidal waveform with
a carrier frequency f, the baseband of the signal
returned from the particle P as a point scatterer is a
function of r(t):
½
¾
2r(t)
s(t) = ½(x, y, z) exp j2¼f
c
= ½(x, y, z) expfj©[r(t)]g
(9)
where ½(x, y, z) is the reflectivity function of the point
scatterer P described in the target local coordinates
(x, y, z), c is the speed of the electromagnetic wave
propagation, and the phase of the baseband signal is
©[r(t)] = 2¼f
2r(t)
:
c
(10)
1 d©(t) 2f d
=
r(t)
2¼ dt
c dt
2f 1 d ~
~ t + <t~r0 )T (R
~0 +V
~ t + <t~r0 )]
[(R + V
c 2r(t) dt 0
¸T
·
2f ~
d
(11)
=
V + (<t~r0 ) ~n
c
dt
=
~ t + <t~r0 )=(kR
~0 + V
~ t + <t~r0 k) is the
~0 + V
where ~n = (R
¡¡!0
unit vector of QP .
To derive the Doppler modulation induced by the
rotation, we utilize a useful relationship ~u £~r = û~r. To
prove it, given a vector ~u = [ux , uy , uz ]T define a skew
symmetric matrix
3
2
0
¡uz uy
7
6
(12)
û = 4 uz
0
¡ux 5 :
¡uy
ux
0
The cross product of the vector ~u and any vector ~r can
be computed through the matrix computation:
2
3 2
32 3
uy rz ¡ uz ry
0
6
7 6
~u £~r = 4 uz rx ¡ ux rz 5 = 4 uz
ux ry ¡ uy rx
¡uy
¡uz
0
ux
uy
rx
0
rz
infinitesimal, and thus (see the Appendix)
ˆ
<t = expf!tg
By taking the time derivative of the phase, the
Doppler frequency shift induced by the target’s
motion is obtained
fD =
Fig. 3. Geometry for radar and vibrating scatterer.
76 7
¡ux 5 4 ry 5 = û~r:
(13)
This relationship is useful in the analysis of the
special orthogonal matrix group or SO(3) rotation
group, also called the 3D rotation matrix [15].
Now we calculate the rotation matrix in (11). In
the reference coordinate system, the angular rotation
velocity vector is described by ~! = (!X , !Y , !Z )T , and
the target will rotate along the unit vector !
~0 = !
~ =k~
!k
with a scalar angular velocity − = k~
! k. Assuming
a high pulse repetition frequency (PRF) and a
relatively low angular velocity, the rotational motion
in each small time interval can be considered to be
(14)
where !ˆ is the skew symmetric matrix associated
with ~!. Thus, the Doppler frequency shift in (11)
becomes
¸T
·
2f ~
2f ~
d !t
ˆ
ˆ
ˆ !t
~r0 )T~n
fD =
V + (e ~r0 ) ~n =
(V + !e
c
dt
c
=
2f ~
2f ~
ˆ r)T~n =
(V + !~
(V + !
~ £~r)T~n:
c
c
(15)
~ t + <t~rk, ~n can be approximated as
~ 0 k À kV
If kR
~ 0 =kR
~ 0 k, which is the direction of the radar
~n = R
LOS. Therefore, the Doppler frequency shift is
approximately
fD =
2f ~
[V + !
~ £~r]radial
c
(16)
where the first term is the Doppler shift due to the
translation and the second term is the micro-Doppler
due to the rotation:
2f
fmicro-Doppler =
(17)
[~! £~r]radial :
c
A. Vibration-Induced Micro-Doppler Modulation
As shown in Fig. 3, the radar is located at the
origin of the radar coordinate system (U, V, W) and
a point scatterer P is vibrating about a center point
O. The center point is also the origin of the reference
coordinate system (X, Y, Z), which is translated from
(U, V, W) at a distance R0 from the radar. We also
assume that the center point O is stationary with
respect to the radar. If the azimuth and elevation angle
of the point O with respect to the radar are ® and ¯,
respectively, the point O is located at
(R0 cos ® cos ¯, R0 sin ® cos ¯, R0 sin ¯)
in the radar coordinates (U, V, W). Then, the unit
vector of the radar LOS becomes
~n = [cos ® cos ¯, sin ® cos ¯, sin ¯]T :
CHEN ET AL.: MICRO-DOPPLER EFFECT IN RADAR: PHENOMENON, MODEL, AND SIMULATION STUDY
(18)
5
Assume that the scatterer P is vibrating at a
frequency fv with an amplitude Dv and that the
azimuth and elevation angle of the vibration direction
in the reference coordinates (X, Y, Z) are ®P and ¯P ,
respectively. As shown in Fig. 3, the vector from
~t =
the radar to the scatterer P becomes then R
~0 + D
~ t and the range of the scatterer P can be
R
expressed as
and thus
µ
¶ X
1
4¼
sR (t) = ½ exp j R0
Jk (B) exp[j(2¼f + k!v )t]
¸
k=¡1
¶
µ
4¼
= ½ exp j R0
¸
£ fJ0 (B) exp(j2¼ft)
+ J1 (B) exp[j(2¼f + !v )t]
~ t j =[(R0 cos ® cos ¯ + Dt cos ®P cos ¯P )2
Rt = jR
¡ J1 (B) exp[j(2¼f ¡ !v )t]
+ (R0 sin ® cos ¯ + Dt sin ®P cos ¯P )2
+ (R0 sin ¯ + Dt sin ¯P )2 ]1=2 :
+ J2 (B) exp[j(2¼f + 2!v )t]
(19)
+ J2 (B) exp[j(2¼f ¡ 2!v )t]
When R0 À Dt , the range is approximately
+ J3 (B) exp[j(2¼f + 3!v )t]
¡ J3 (B) exp[j(2¼f ¡ 3!v )t] + ¢ ¢ ¢ g:
Rt = fR02 + Dt2 + 2R0 Dt [cos(® ¡ ®p ) cos ¯ cos ¯p + sin ¯ sin ¯p ]g1=2
¼ R0 + Dt [cos(® ¡ ®p ) cos ¯ cos ¯p + sin ¯ sin ¯p ]:
(20)
If the azimuth angle ® of the center point O and
the elevation angle ¯P of the scatterer P are all zero,
when R0 À Dt we have
Rt = (R02 + Dt2 + 2R0 Dt cos ®P cos ¯)1=2
»
= R0 + Dt cos ®P cos ¯:
(21)
Because the vibration rate of the scatterer in angular
frequency is !v = 2¼fv and the amplitude of the
vibration is Dv , Dt = Dv sin !v t and the range of the
scatterer becomes
R(t) = Rt = R0 + Dv sin !v t cos ®P cos ¯:
(22)
Z
sin ¯P
Z0
(27)
The velocity of the scatterer P due to the vibration
becomes
(28)
Based on (16), the micro-Doppler shift induced by
the vibration is
(23)
where ½ is the reflectivity of the point scatterer, f
is the carrier frequency of the transmitted signal, ¸
is the wavelength, and ©(t) = 4¼R(t)=¸ is the phase
modulation function.
Substituting (22) into (23) and denoting B =
(4¼=¸)Dv cos ®P cos ¯, the received signal can be
rewritten as
¾
½
4¼
sR (t) = ½exp j R0 expfj2¼ft + B sin !v tg
¸
(24)
which can be further expressed by the Bessel function
of the first kind of order k:
Z ¼
1
Jk (B) =
expfj(B sin u ¡ ku)du
(25)
2¼ ¡¼
6
Therefore, the micro-Doppler frequency spectrum
consists of pairs of spectral lines around the center
frequency f with spacing !v =(2¼) between adjacent
lines.
Because of the vibration, the point scatterer P,
which initially at time t = 0 is located at (X0 , Y0 , Z0 )T
in the (X, Y, Z) coordinates, will, at time t, move to
3 2 3
2
2 3
X0
cos ®P cos ¯P
X
7 6 7
6
6 7
4 Y 5 = Dv sin(2¼fv t) 4 sin ®P cos ¯P 5 + 4 Y0 5 :
~v = 2¼Dv fv cos(2¼fv t)(cos ®P cos ¯P , sin ®P cos ¯P , sin ¯P )T :
The radar received signal becomes then
½ ·
¸¾
R(t)
sR (t) = ½ exp j 2¼ft + 4¼
¸
= ½ expfj[2¼ft + ©(t)]g
(26)
2f T
(~v ¢ ~n)
c
4¼ffv Dv
=
[cos(® ¡ ®P ) cos ¯ cos ¯P
c
+ sin ¯ sin ¯P ] cos(2¼fv t):
fmicro-Doppler =
(29)
If the azimuth angle ® and the elevation angle ¯P are
both zero, one has
fmicro-Doppler =
4¼ffv Dv
cos ®P cos ¯ cos(2¼fv t):
c
(30)
When the orientation of the vibrating scatterer is along
the projection of the radar LOS direction, or ®P = 0,
and the elevation angle ¯ is also 0, the Doppler
frequency change reaches the maximum value of
4¼ffv Dv =c.
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where
a11 = cos Á cos à ¡ sin Á cos µ sin Ã
a12 = ¡ cos Á sin à ¡ sin Á cos µ cos Ã
a13 = sin Á sin µ
a21 = sin Á cos à + cos Á cos µ sin Ã
a22 = ¡ sin Á sin à + cos Á cos µ cos Ã
a23 = ¡ cos Á sin µ
Fig. 4. Geometry of radar and rotating target.
a31 = sin µ sin Ã
a32 = sin µ cos Ã
a33 = cos µ:
Fig. 5. Rotations by Euler angles (Á, µ, Ã).
B. Rotation-Induced Micro-Doppler Modulation
The geometry of a radar and a target with
3-dimensional rotations is illustrated in Fig. 4. The
radar coordinate system is (U, V, W), the target
local coordinate system is (x, y, z), and the reference
coordinate system (X, Y, Z) is parallel to the radar
coordinates (U, V, W) and located at the origin of the
target local coordinates. Assume that the azimuth and
elevation angle of the target in the radar coordinates
(U, V, W) are ® and ¯, respectively, and the unit vector
of the radar LOS is the same as (18).
Due to the target’s rotation, any point on the target
described in the local coordinate system (x, y, z) will
move to a new position in the reference coordinate
system (X, Y, Z). The new position can be calculated
from its initial position vector multiplied by an initial
rotation matrix determined by Euler angles (Á, µ, Ã),
where the angle Á rotates about the z-axis, the angle µ
rotates about the x-axis, and the angle à rotates about
the z-axis again, as illustrated in Fig. 5.
The corresponding initial rotation matrix is defined
by
2
3
a11 a12 a13
6
7
<Init = 4 a21 a22 a23 5
2
a31
a32
a33
32
1
0
cos Á ¡ sin Á 0
76
6
= 4 sin Á cos Á 0 5 4 0 cos µ
2
0
0
cos Ã
6
£ 4 sin Ã
0
1
¡ sin Ã
0
0
1
cos Ã
0
3
7
05
sin µ
0
3
7
¡ sin µ 5
cos µ
(31)
Viewed in the target local coordinate system,
when a target rotates about its axes x, y, and z with
an angular velocity !
~ = (!x , !y , !z )T , a point scatterer
T
P at ~r0 = (x0 , y0 , z0 ) represented in the target local
coordinates (x, y, z) will move to a new location in the
reference coordinate system described by <Init ¢~r0 and
the unit vector of the rotation becomes
!
~ 0 = (!x0 , !y0 , !z0 )T =
~
<Init ¢ !
k~! k
(32)
with the scalar angular velocity − = k~! k. Thus,
according to Rodrigues formula (see the Appendix)
[15], at time t the rotation matrix becomes
<t = I + !ˆ 0 sin −t + !ˆ 02 (1 ¡ cos −t)
where !ˆ 0 is a skew symmetric matrix
3
2
0
¡!z0
!y0
7
6
!ˆ 0 = 4 !z0
0
¡!x0 5 :
¡!y0
!x0
(33)
(34)
0
Viewed in the reference coordinate system
(X, Y, Z), at time t, the scatterer P will move from
its initial location to a new location ~r = <t ¢ <Init ¢~r0 .
According to (17), the micro-Doppler frequency shift
induced by the rotation is approximately
fmicro-Doppler =
=
=
2f
2f
[−~! 0 £~r]radial =
(− !ˆ 0~r)T ¢ ~n
c
c
2f
[− !ˆ 0 <t ¢ <Init ¢~r0 ]T ¢ ~n
c
2f−
f[!ˆ 02 sin −t ¡ !ˆ 03 cos −t
c
+ !ˆ 0 (I + !ˆ 02 )]<Init ¢~r0 gT ¢ ~n:
(35)
If the skew symmetric matrix !ˆ 0 is defined by a
unit vector, then !ˆ 03 = ¡!ˆ 0 and the rotation-induced
CHEN ET AL.: MICRO-DOPPLER EFFECT IN RADAR: PHENOMENON, MODEL, AND SIMULATION STUDY
7
Fig. 7. Frequency representation of signal reflected from
vibrating scatterer.
Fig. 6. Geometry of radar platform vibration.
micro-Doppler frequency becomes
fmicro-Doppler =
2f− 0 0
[!ˆ (!ˆ sin −t + I cos −t)<Init ¢~r0 ]radial :
c
(36)
C. Effect of Radar Platform Vibration on
Micro-Doppler Modulation
If the radar platform is also vibrating, the radar
can be moved from the origin of the coordinate
system (U, V, W) at a distance DR as shown in Fig. 6.
Thus, the vector from the radar to a scatterer P on the
~t = R
~0 + D
~t ¡D
~ R where D
~ R represents
target becomes R
the component of the platform’s vibration. Thus, if
the azimuth angle ® of the center point O is zero, the
range from the radar to the scatterer P becomes
R(t) = R0 + Dv sin !v t ¢ (cos ®P cos ¯P cos ¯ + sin ¯P sin ¯)
¡ DR sin !R t ¢ (cos ®0 cos ¯0 cos ¯ + sin ¯0 sin ¯)
(37)
where !R is the platform vibration rate in angular
frequency, DR is the amplitude of the platform
vibration, and ®0 and ¯0 are the azimuth and elevation
angles of the direction of radar platform vibration
with respect to the initial radar origin point.
If the platform vibration and the target vibration
are in the same direction (®0 = ®P and ¯0 = ¯P ),
the vibration of the radar platform is superimposed
onto the target’s vibration. Otherwise, the platform
vibration may have a more complex effect on the
micro-Doppler modulation.
The micro-Doppler modulation induced by these
vibrations becomes
fmicro-Doppler
=
1 d© 2 dR(t)
=
2¼ dt
¸ dt
=
2
D ! [cos(® ¡ ®p ) cos ¯ cos ¯p + sin ¯ sin ¯p ] cos !v t
¸ v v
¡
2
D ! [cos(® ¡ ®0 ) cos ¯0 cos ¯ + sin ¯0 sin ¯] cos !R t:
¸ R R
(38)
8
The second term is the micro-Doppler variation
caused by the platform vibration. When ® = ®P = ®0
and ¯ = ¯P = ¯0 = 0, the micro-Doppler modulation
becomes
2
¢fmicro-Doppler = (Dv !v cos !v t ¡ DR !R cos !R t):
¸
(39)
If the radar platform vibrates at a lower rate, for
eliminating the variation caused by the platform
vibration, a low-pass filter may be used to extract
and remove the second micro-Doppler modulation
term. The platform vibration can also be filtered out
in the joint time-frequency domain by applying a
time-frequency filtering.
III. TIME-FREQUENCY ANALYSIS OF
MICRO-DOPPLER MODULATION
The Fourier transform is the most common method
to analyze the properties of a signal waveform in
the frequency domain. It shows the distribution of
the magnitude and the phase at different frequencies
contained in the signal during the time interval
of analyzing. When a signal is reflected from
a rotating target, the frequency spectrum of the
signal may indicate the presence of micro-Doppler
modulation. Micro-Doppler effect can be observed by
deviations of the frequency spectrum from the center
frequency. Fig. 7 shows the frequency spectrum of
a signal reflected by a vibrating point scatterer. The
micro-Doppler frequencies are centered at 0 with an
equal displacement toward each side, suggesting that
the point scatterer could have been vibrating along
the radar LOS with equal magnitude toward each
direction.
Due to lack of localized time information, the
Fourier transform, however, cannot provide more
complicated time-varying frequency modulation
information. A joint time-frequency analysis that
provides localized time-dependent frequency
information is needed for extracting time-varying
motion dynamic features.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 1
JANUARY 2006
TABLE I
Bilinear Time-Frequency Transforms
TF Transforms
Cohen Class
Formula
C(t, !) =
Wigner-Ville
RRR
Choi-Williams
CW(t, !, ¾) =
Pseudo Wigner
RR
R ³
¿ ¤
¿
s t+
2
2
Cone Kernel
CKD(t, !) =
RRR
´ ³
s t+
³
R
³
h(¿ )s t +
SPWD(t, !, ®) =
Wigner-Ville (SPWV)
´ ³
KCW (u ¡ t, ¿ )s u +
PWD(t, !, ®) =
Smooth Pseudo
¿ ¤
¿
s u+
2
2
©(µ, ¿ )s u +
WVD(t, !) =
Kernel
³
R
´
e¡jµt¡j!¿ +jµu du d¿ dµ
´
´ ³
¿ ¤
¿
s t+
2
2
´
³
´ ³
(40)
¤
0
e¡j!¿ du d¿
expf¡j!¿ gd¿
¿ ¤
¿
s u+
2
2
To analyze the time-varying frequency
characteristics of the micro-Doppler modulation
and visualize the localized joint time and frequency
information, the signal must be analyzed by using
a high-resolution time-frequency transform, which
characterizes the temporal and spectral behavior of the
analyzed signal. For example, by examining the time
information and the sign of the micro-Doppler shift
caused by a movement, the direction of the movement
at the specific time could be found.
Time-frequency transforms includes linear
transforms, such as the short-time Fourier transform
(STFT), and bilinear transforms, such as the
Wigner-Ville distribution (WVD). With a time-limited
window function, the resolution of the STFT
is determined by the window size. There is a
trade-off between the time resolution and the
frequency resolution. A larger window has higher
frequency resolution but a poorer time resolution.
The well-known spectrogram defined as the
square modulus of the STFT is a popular tool for
time-frequency analysis.
The WVD of a signal s(t) is defined as the Fourier
transform of the time-dependent autocorrelation
function
¶ µ
¶
Z µ
t0 ¤
t0
WVD(t, !) = s t +
s t¡
expf¡j!t0 gdt0
2
2
0
´
where s(t + (t =2))s (t ¡ (t =2)) can be seen as a
time-dependent autocorrelation function. The bilinear
WVD has better joint time-frequency resolution than
any linear transform. It suffers, however, from a
problem of cross-term interference, i.e., the WVD
of the sum of two signals is not the sum of their
individual WVDs. If a signal contains more than one
component in the joint time-frequency domain, its
WVD will contain cross terms that occur halfway
´
©(µ, ¿ ) = KCW (µ, ¿ ) =
1
4¼ 3=2
p
expf¡µ2 ¿ 2 =¾g
¿ 2 =¾
©(µ, ¿ ) = h(¿ ) = expfj®¿ 2 =2g
©(µ, ¿ ) = h(¿ ) = expfj®¿ 2 =2g
s(t ¡ u)PWD(t, !, ®)du
KCK (t ¡ u, ¿ )s u +
©(µ, ¿ ) = 1
expf¡j!¿ gd¿
¿ ¤
¿
s u+
2
2
´ ³
©(µ, ¿ )
e¡j!¿ du d¿
©(µ, ¿ ) = KCK (t, ¿ ) =
½
g(¿ );
jt=¿ j < 1=2
0;
jt=¿ j > 1=2
between each pair of auto-terms. The magnitude of
these oscillatory cross terms can be twice as large as
the auto-terms.
To reduce the cross-term interference, the filtered
WVD has been used to preserve the useful properties
of the time-frequency transform with a slightly
reduced time-frequency resolution and largely reduced
cross-term interference. The WVD with a linear
low-pass filter belongs to Cohen’s class [18, 19].
The general form of Cohen’s class is defined as
ZZ ³
¿ ´ ¤³
¿´
C(t, !) =
s u+
s u¡
2
2
£ Á(t ¡ u, ¿ ) expf¡j!¿ gdu d¿:
(41)
The Fourier transform of Á(t, ¿ ), denoted as ©(µ, ¿ ), is
called the kernel function. It can easily be seen that
if ©(µ, ¿ ) = 1, then Á(t, ¿ ) = ±(t) and the Cohen class
reduces to the WVD. Other types of kernel functions,
which lead to the Choi-Williams distribution, the
pseudo Wigner, the smoothed pseudo Wigner-Ville
and the cone kernel distribution (see Table I), can
be designed to reduce the cross-term interference
problem of the WVD.
Other high-resolution time-frequency transform
includes the adaptive time-frequency transform [18]. It
decomposes a signal into a family of basis functions,
such as the Gabor function (a Gaussian modulated
exponential function), which is well localized in both
the time and the frequency domain and adaptive to
match the local behavior of the analyzed signal. The
adaptive Gaussian representation is a signal-adaptive,
high-resolution transform. It expands a signal s(t)
in terms of Gabor basis functions hp (t) with an
adjustable standard deviation sp and a time-frequency
center (tp , fp ):
1
X
s(t) =
Bp hp (t)
(42)
p=1
CHEN ET AL.: MICRO-DOPPLER EFFECT IN RADAR: PHENOMENON, MODEL, AND SIMULATION STUDY
9
Fig. 8. Block diagram of simulation.
where
"
hp (t) = (¼¾p2 )¡0:25 exp ¡
of a walking man) using their different components
along the basis functions.
#
(t ¡ tp )2
exp(j2¼fp t):
2¾p2
(43)
The coefficients Bp are found by an iterative
procedure beginning with the stage p = 1 and
choosing the parameters ¾p , tp , and fp such that hp (t)
is most similar to s(t):
¯Z
¯2
¯
¯
2
¤
¯
jBp j = max ¯ sp¡1 (t)hp (t)dt¯¯
(44)
¾p ,tp ,fp
where s0 (t) = s(t), i.e., the analyzed signal is taken
as the initial signal for p = 1. For p > 1, sp (t) is the
residual after the orthogonal projection of sp¡1 (t) onto
hp (t) has been removed from the signal:
sp (t) = sp¡1 (t) ¡ Bp (t)hp (t):
(45)
This procedure is iterated to generate as many
coefficients as needed to accurately represent the
original signal.
In principle, any time-frequency transform can be
used to analyze micro-Doppler modulations. However,
a desired time-frequency transform should satisfy the
requirements on high resolution in both the time and
frequency domains and low cross-term interference.
The smoothed pseudo Wigner-Ville distribution
in Table I is a good candidate for analyzing
micro-Doppler modulations due to its slightly reduced
time-frequency resolution and largely reduced
cross-term interference. The adaptive time-frequency
transform is also a good candidate and has advantages
over conventional time-frequency techniques (such
as STFT). It is a parametric procedure that not only
results in very high time-frequency resolution but also
allows to separate different target’s features (such as
the swinging arm feature and the body motion feature
10
IV. SIMULATION STUDY OF MICRO-DOPPLER
MODULATIONS INDUCED BY MICRO-MOTION
DYNAMICS
In this section, we present several examples
of micro-motions that can induce micro-Doppler
modulations. Based on the basic model described in
Section II, several useful formulas of micro-Doppler
modulations are derived. We compare the theoretical
results with those generated by computer simulation.
In the simulation, the target is defined in terms of
a 3D reflectivity density function characterized by
point scatterers. No occlusion effect is considered in
target models. We select the point scatterer model
in the simulation study because of its simplicity
compared with electromagnetic prediction codes
(such as the Xpatch), its ability to incorporate
any target’s motion into the simulation, and its
capability to investigate the effect of individual motion
components. In both the simulated micro-Doppler
modulations and the theoretical micro-Doppler
modulations calculated from the corresponding
formulas, we use the same target model with the same
micro motions. The simulation, however, does not
presuppose any theoretical formulas of micro-Doppler
modulations. It is based on physical properties of
electromagnetic backscattering from a rigid body
undergoing nonlinear dynamics. The purpose of the
simulation is to verify the theoretical results. The
comparison shows that the simulation result and the
theoretical result are identical, and thus verifies the
theoretical formulation of micro-Doppler modulations.
Fig. 8 is the block diagram of the simulation.
The system mainly consists of three parts: the radar
transmitted signal, the target, and the radar received
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 1
JANUARY 2006
signal. The transmitted signal can be a step-frequency
waveform, an impulse waveform, a Doppler chirp
waveform, or other defined waveforms. The target is
given by a set of point scatterers described by their
reflectivity and locations in the target local coordinate
system (x, y, z) embedded on the target, their origin
at the geometric center while the x-axis is defined
as the heading direction of the target. The radar is
located at the origin of the radar coordinates (U, V, W).
Both the radar and the target coordinates can have
translations and rotations with respect to the Earth
center Earth fixed (ECEF) system. Initial locations,
velocities, accelerations and trajectories of the radar
and the target coordinates are predetermined as inputs
to the simulation.
Based on the returned signal from a single point
scatterer, the returned signal from the target can be
represented as the summation of the returned signals
from all scatterer centers in the target. For a given
PRF, the transmitter repeatedly transmits signals.
For each transmitted signal, the returned signal from
the target must be updated according to the updated
range for each point scatterer on the target due to its
translation and rotations. Then, the returned signals
can be rearranged into a two-dimensional (the number
of range cells by the number of pulses) data matrix,
which is the raw data to be analyzed. In summary, the
simulation procedure includes [19]:
1) select radar parameters (the carrier frequency,
bandwidth, signal waveform, and PRF);
2) select radar motion parameters (the initial
location, trajectory, velocity, and acceleration in the
ECEF system);
3) select predesigned point scatterer model of the
target;
4) select target motion parameters (the initial
location, trajectory, velocity, and acceleration in the
ECEF system);
5) transmit signal pulse repeatedly to the target,
update the radar and the target locations, and calculate
received signals;
6) select signal-to-noise ratio (SNR) for the raw
data and add the calculated noise level to the raw data;
7) arrange the received raw data into a
2-dimensional (range £ pulse) matrix.
In the simulation procedure, no mathematical formula
on micro-Doppler modulations is implemented.
A. Vibrating Target
Vibration is a basic micro-motion of a target.
Examples of vibrations include engine-induced
surface vibration and mechanical oscillations of a
bridge or a building. The geometry of the radar and
a vibrating point scatterer is shown in Fig. 9(a). The
vibration center O is stationary with azimuth angle ®
and elevation angle ¯ with respect to the radar. The
scatterer is vibrating at a vibration frequency fv and
amplitude Dv . The azimuth and elevation angle of
the vibrating direction are ®P and ¯P , respectively,
described in the reference coordinates (X, Y, Z). Then,
with the unit vector of the radar LOS given in (18),
the point scatterer P, which is initially located at time
t = 0 at (X0 , Y0 , Z0 ) in (X, Y, Z), will move at time t to
3 2 3
2
2 3
X0
cos ®P cos ¯P
X
7 6 7
6
6 7
4 Y 5 = Dv sin(2¼fv t) 4 sin ®P cos ¯P 5 + 4 Y0 5
Z
sin ¯P
Z0
(46)
as indicated in (27). The micro-Doppler modulation
induced by the vibration is derived in (29)
fmicro-Doppler
=
4¼ffv Dv
[cos(® ¡ ®P ) cos ¯ cos ¯P + sin ¯ sin ¯P ] cos(2¼fv t):
c
(47)
Assume the radar operates at 10 GHz and
transmits a pulse waveform with a PRF of 2,000.
Given Dv = 0:01 m, fv = 2 Hz, ®P = 30± , ¯P = 30± ,
and the center of the vibration at (U = 1000 m,
V = 5000 m, W = 5000 m), the theoretical result of
the micro-Doppler modulation is shown in Fig. 9(b),
while the simulation result shown in Fig. 9(c) is
identical to the theoretical analysis.
B. Rotating Target
Rotation is another basic micro-motion of a
target. Examples of rotations include rotor blades
of a helicopter or a rotating antenna on a ship. The
geometry of the radar and a 3-dimensional rotating
target is depicted in Fig. 10(a). The radar coordinate
system is defined by (U, V, W); the target local
coordinate system is defined by (x, y, z); and the
reference coordinate system (X, Y, Z) is parallel to
the radar coordinates (U, V, W) and located at the
origin of the target local coordinates. The azimuth and
elevation angle of the target with respect to the radar
coordinates (U, V, W) are ® and ¯, respectively.
Assume the target rotates about its axes x, y, and
z with an angular velocity vector ~! = (!x , !y , !z )T or
a scalar angular velocity − = k~! k. The corresponding
initial rotation matrix <Init is given in (31). If a point
scatterer P is initially located at ~r0 = (x0 , y0 , z0 )T in
the local coordinates (x, y, z), then viewing from
the reference coordinates system (X, Y, Z) the point
scatterer P will move to <Init ¢~r0 through rotating
along the unit vector
~ 0 = (!x0 , !y0 , !z0 )T
!
given in (32). Therefore, reviewing in the reference
coordinate system (X, Y, Z), at time t the scatterer P
CHEN ET AL.: MICRO-DOPPLER EFFECT IN RADAR: PHENOMENON, MODEL, AND SIMULATION STUDY
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Fig. 9. Micro-Doppler modulation induced by vibration.
Fig. 10. Micro-Doppler modulation induced by rotation.
will move to ~r = <t ¢ <Init ¢~r0 . According to (35), if
!ˆ 03 = ¡!ˆ 0 , the micro-Doppler modulation of the point
scatterer becomes
fmicro-Doppler =
2f− 0 0
[!ˆ (!ˆ sin −t + I cos −t)<Init ¢~r0 ]radial
c
as given in (36), and the micro-Doppler signature in
the time-frequency domain is a sinusoidal function of
− with an initial phase and amplitude depending on
the initial position and the initial Euler angles (Á, µ, Ã)
of the point scatterer.
Assume that the radar operates at 10 GHz and a
target, located at (U = 1000 m, V = 5000 m, W =
5000 m), is rotating along the x, y, and z axes with
12
an initial Euler angles (Á = 30± , µ = 20± , Ã = 20± )
and angular velocity !
~ = [¼, 2¼, ¼]T rad/s. Suppose
the target has three strong scatterer centers: scatterer
P0 (the center of the rotation) is located at (x = 0 m,
y = 0 m, z = 0 m); scatterer P1 is located at (x = 1:0 m,
y = 0:6 m, z = 0:8 m); and scatterer P2 is located
at (x = ¡1:0 m, y = ¡0:6 m, z = ¡0:8 m). The
theoretical micro-Doppler modulation calculated by
(36) is shown in Fig. 10(b). The micro-Doppler of
the center point P0 is the line at the zero frequency,
and micro-Doppler modulations from the point P1
and P2 are the two sinusoidal curves about the zero
frequency. Given a radar PRF of 1,000 pulse/s and
2,048 pulses transmitted during 2.05 s of dwell time,
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 1
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point scatterer P1 will move to <Init ¢~r0 and the initial
~ = V ¢ [a11 , a21 , a31 ]T ,
velocity of the target will be V
where a11 , a21 , and a31 are defined in (31).
Suppose the target rotates along the y-axis with an
angular velocity ! = !y rad/s. The trajectories of the
target center of mass O and the scatterer P1 are plotted
in Fig. 11(b), where the dot-line is the trajectory of
the target’s center and the circled-line is the trajectory
of the scatterer P1 . During tumbling, the length of
the segment OP1 stays constant. Viewed from the
reference coordinates (X, Y, Z), the angular velocity
of the target is !
~ 0 = ! ¢ [a12 , a22 , a32 ]T and the location
of the scatterer P1 will move to
3
2
cos !t 0 sin !t
7
6
~r = <Init 4 0
(48)
1
0 5~r0 :
¡ sin !t
Fig. 11. (a) Geometry of radar and tumbling target.
(b) Trajectory of scatterer in tumbling target.
the simulated micro-Doppler modulation signature
is shown in the Fig. 10(c), which is identical to
the theoretical analysis. The rotation period can
be obtained from the rotation angular velocity as
T = 2¼=k~
!k = 0:8165 s and is verified by both the
theoretical result and the simulated result.
C. Tumbling Target
Tumbling is a rotation accompanied by translation
and acceleration. Examples of target tumbling include
a spacecraft tumbling through space or a fuel tank of
a missile tumbling after its separation from the reentry
vehicle. Tumbling target with translation, acceleration,
and rotation is illustrated in Fig. 11(a). Assume the
~ along the
target has an initial translation velocity V
x-axis and an acceleration of g = 9:8 m/s2 , due to
gravity, with respect to the radar coordinates. At the
same time, the target rotates along the y-axis with an
angular velocity !y . The azimuth and elevation angle
of the origin O of the target local coordinates with
respect to the origin of the radar coordinates are ®
and ¯, respectively. The reference coordinate system
(X, Y, Z), which is parallel to the radar coordinates
(U, V, W) and shares the same origin O with the target
local coordinates (x, y, z), has the same initial velocity
and acceleration as the target but has no rotation with
respect to the radar coordinates. Thus, the location
of any point scatterer of the target in (X, Y, Z) can be
calculated by multiplying the location of the scatterer
in (x, y, z) with an initial rotation matrix <Init defined
by the initial Euler angles (Á, µ, Ã) given in (31).
Assuming the initial location of a scatterer P1 is at
~r0 = [x, y, z]T in the local coordinate system, viewed
from the reference coordinate system (X, Y, Z) the
0 cos !t
Thus, the velocity of the target at time t becomes
~ = [Va11 , Va21 , Va31 ¡ gt]T . According to the
V
mathematical formula in (16), the Doppler frequency
induced by the tumbling becomes
fD =
=
2f ~
[V + ~!0 £~r]radial
c
2f
(¡ sin ¯gt + t1 )
c
+
2f!
[(xt2 + zt3 ) cos !t + (zt2 ¡ xt3 ) sin !t]
c
(49)
where
t1 = V[(a11 cos ® + a21 sin ®) cos ¯ + a31 sin ¯]
t2 = [(a22 a31 ¡ a32 a21 ) cos ® + (a32 a11 ¡ a12 a31 ) sin ®] cos ¯
+ (a12 a21 ¡ a22 a11 ) sin ¯
(50)
t3 = [(a22 a33 ¡ a32 a23 ) cos ® + (a32 a13 ¡ a12 a33 ) sin ®] cos ¯
+ (a12 a23 ¡ a22 a13 ) sin ¯:
The Doppler frequency shift induced by the initial
velocity is
2f
fDTrans =
(51)
t
c 1
and the micro-Doppler modulation induced by the
rotation is
2f!
fmicro-Doppler =
[(xt2 + zt3 ) cos !t + (zt2 ¡ xt3 ) sin !t]
c
2f
¡
sin ¯ ¢ gt:
(52)
c
We notice that the micro-Doppler modulation induced
by tumbling has a similar format as the micro-Doppler
of rotation except an additional linear time-varying
term determined by the gravity acceleration.
Assume now that the radar operates at 10 GHz and
a target located at (U = 1000 m, V = 5000 m, W =
5000 m) is rotating along the y-axis with an angular
CHEN ET AL.: MICRO-DOPPLER EFFECT IN RADAR: PHENOMENON, MODEL, AND SIMULATION STUDY
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Fig. 12. Micro-Doppler modulations induced by target’s tumbling.
velocity !y = 4¼ rad/s with initial Euler angles of
Á = 20± , µ = 20± , Ã = 20± . If the initial velocity of the
target is 5 m/s, the initial position of the scatterer P0 is
at the origin (x = 0 m, y = 0 m, z = 0 m), the scatterer
P1 is at (x = ¡0:5 m, y = 0:3 m, z = 0:4 m), and the
scatterer P2 is at (x = 0:5 m, y = ¡0:3 m, z = ¡0:4 m),
then the theoretical micro-Doppler modulation is
calculated by (52) and shown in Fig. 12(a). Due
to gravity, the micro-Doppler of the point P0 is a
slope line starting at 200 Hz, and micro-Doppler
modulations from the point P1 and P2 are the two
sinusoidal curves around the slope line. For computer
simulation, assuming the PRF is 3,000 pulse/s,
i.e., 3,000 pulses can be transmitted within 1 s of
dwell time, the simulated micro-Doppler modulation
signature in the joint time-frequency domain is shown
in Fig. 12(b), which is identical to the theoretical
result. The rotation period can be obtained from
the rotation angular velocity as T = 2¼=k~!k = 0:5 s
and is verified by both the theoretical result and the
simulated result.
D. Coning Target
~n = [cos ® cos ¯, sin ® cos ¯, sin ¯]T
as given in (18), and the unit vector of the rotation
axis in the reference coordinates (X, Y, Z) is
~e = [cos ®N cos ¯N , sin ®N cos ¯N , sin ¯N ]T :
Coning motion is a rigid body rotation about
an axis that intersects with an axis of the local
coordinates. An example of the coning motion is the
conning of a whipping top, which is a symmetric
body spinning around its axis of symmetry and with
one end point fixed. When its axis of symmetry is
rotating about an axis that intersects with the axis of
symmetry, the top is also doing coning motion. If the
axis of symmetry does not remain at a constant angle
with the axis of coning, it will oscillate up and down
between two limits. This motion is called nutation.
While a fuel tank of a missile is tumbling after its
separation from the reentry vehicle as mentioned in
Section IVC, the reentry vehicle will have spinning,
coning, and nutation motions.
14
For simplicity, we only deal with target’s coning
motion without spinning and nutation. The geometry
of the radar and a target with coning motion is
depicted in Fig. 13(a). The radar is located at the
origin of the radar coordinate system (U, V, W)
and the target’s local coordinate system is (x, y, z).
¡!
The target has a coning motion along the axis SN,
which intersects with the z-axis at the point S (x = 0,
y = 0, z = z0 ) of the local coordinates. The reference
coordinate system (X, Y, Z) is parallel to the radar
coordinates (U, V, W) and its origin is located at
the point S. Assume the azimuth and elevation
angle of the target center O with respect to the
radar are ® and ¯, respectively, and the azimuth and
¡!
elevation angle of the coning axis SN with respect
to the reference coordinates (X, Y, Z) are ®N and
¯N , respectively. Then, the unit vector of the radar
LOS is
(53)
Assume the initial position of a scatterer P1 is ~r0 =
[x, y, z]T represented in the target local coordinates.
Then, the location of the point scatterer P1 in the
reference coordinates (X, Y, Z) can be calculated
through its local coordinates by subtracting the
coordinates of the point S and multiplying the rotation
matrix <Init defined by the initial Euler angles (Á, µ, Ã).
Viewed from the reference coordinates (X, Y, Z),
the location of the scatterer P1 is at <Init ¢ [x, y, z ¡ z0 ]T .
Suppose the target has a coning motion with an
angular velocity of ! rad/s. According to the
Rodrigues’s formula, at time t the rotation matrix in
the (X, Y, Z) becomes
<t = I + ê sin !t + ê2 (1 ¡ cos !t)
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 1
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Fig. 13. (a) Geometry of radar and target with coning motion. (b) Calculated. (c) Simulated signatures of micro-Doppler modulation
induced by coning motion.
where the skew symmetric matrix is defined by
2
6
ê = 4
0
¡ sin ¯N
sin ¯N
0
¡ sin ®N cos ¯N
cos ®N cos ¯N
sin ®N cos ¯N
3
7
¡ cos ®N cos ¯N 5 :
0
(55)
Therefore, at time t the scatterer P1 will move to
~r(t) = <t ¢ <Init ¢ [x, y, z ¡ z0 ]T .
If the point S is not too far from the target center
of mass, we can approximate the radar LOS as the
radial direction of the point S with respect to the
radar. According to the mathematical formula (11)
described in Section II, the micro-Doppler modulation
induced by the coning motion is approximately
fmicro-Doppler
2f
=
c
=
·
d
~r(t)
dt
¸
2f
=
c
radial
½·
¸
d
< < ¢ [x, y, z ¡ z0 ]T
dt t Init
¾T
¢ ~n
2f!
f(ê cos !t + ê2 sin !t)<Init ¢ [x, y, z ¡ z0 ]T gT ¢ ~n:
c
(56)
Assume that the radar operates at 10 GHz and a
target is initially located at (U = 1000 m, V = 5000 m,
W = 5000 m). Suppose the initial Euler angles are
Á = 30± , µ = 30± , and à = 45± , the target is coning
with an angular velocity ! = 4¼ rad/s, and the azimuth
and elevation angle of the rotation axis are ®N = 60±
and ¯P = 45± , respectively. Thus, given the location
of the point S at (x = 0 m, y = 0 m, z = ¡1 m), the
initial location of the scatterer P0 at the point S, the
scatterer P1 at (x = 0:5 m, y = 0:6 m, z = 1:0 m),
and the scatterer P2 at (x = ¡0:5 m, y = ¡0:6 m,
z = ¡1:0 m), then the theoretical micro-Doppler
modulation calculated by (56) is shown in Fig. 13(b).
The micro-Doppler of the point P0 is zero frequency,
and the micro-Doppler modulations from points
P1 and P2 are the two sinusoidal curves about the
zero frequency. For computer simulation, assuming
the PRF is 2,000 pulse/s, i.e., 2,000 pulses can be
transmitted within 1 s of dwell time, the simulated
micro-Doppler modulation signature in the joint
time-frequency domain is shown in Fig. 13(c), which
is identical to the theoretical result and verifies that
the rotation period is T = 2¼=k~!k = 0:5 s.
V.
APPLICATIONS OF MICRO-DOPPLER ANALYSIS
We have discussed the micro-Doppler effect in
radar, joint time-frequency analysis of micro-Doppler
modulations, and the simulation of micro-Doppler
modulations using the point scattering model. In this
section, we demonstrate the micro-Doppler effect
in radar using measured real radar data and analyze
micro-Doppler features embedded in radar signals
using time-frequency analysis.
Radar signals returned from a rotating antenna
on a ship, propellers of a fixed-wing aircraft, a rotor
of a helicopter, or an engine compressor and blade
assemblies of a jet aircraft, contain micro-Doppler
characteristics related to the structures’ rotation that
can be used to determine the motion dynamics of
these structures. Vibration generated by a vehicle
engine can be detected by radar signals returned
from the surface of the vehicle. Micro-Doppler
signature of the engine vibration signal can be used
to identify specific type of vehicles and determine
CHEN ET AL.: MICRO-DOPPLER EFFECT IN RADAR: PHENOMENON, MODEL, AND SIMULATION STUDY
15
Fig. 14. Micro-Doppler signatures of vibrating cornel reflectors.
the movement and the speed of the engine. Natural
mechanical oscillations of a bridge or a building can
be also detected by the micro-Doppler modulations
in radar returned signals. In Section VA, we analyze
micro-Doppler signatures of two vibrating corner
reflectors on the ground using measured real synthetic
aperture radar data.
Human identification (ID) at a distance is the
subject of increasing interest in biometrics and
computer vision. Gait is a “gesture” of a walking
person distinguishable from person to person. Gait
analysis is a method of analyzing human motion
from distance that offers a new way for human ID.
A person will usually perform the same unique pattern
when walking. It should be thus possible to recognize
a person at a distance from their gait. Most efforts
on human gait analysis are using image sequences.
Here, in Section VB, we describe the micro-Doppler
signature of a walking person using real radar data
returned from the walking person.
A. Micro-Doppler Signature of Vibrating Corner
Reflectors in SAR Scene
In synthetic aperture radar (SAR) image,
oscillating targets may cause phase modulation on the
azimuth phase history data. This phase modulation
can be seen as a time-dependent micro-Doppler
frequency shift. Based on micro-Doppler signatures in
the time-frequency domain, references [20] and [21]
calculated the oscillating frequency and amplitude of
vibrating corner reflectors in a SAR scene.
Fig. 14(a) is a part of a SAR scene taken by
X-band airborne SAR operating at 9.6 GHz with a
PRF of 1,000 Hz. There are two corner reflectors
16
on the ground as shown in Fig. 14(b). The reflectors
were mechanically vibrating when the airborne
SAR was illuminating the scene. As indicated in
the zoomed area of Fig. 14(a), the two vibrating
reflectors are located at range cell 393 and 400 in
the SAR image. Due to vibration, SAR images of
the two reflectors are smeared along the cross-range
direction. In order to extract micro-Doppler signatures
of the two vibrating reflectors, the phase history
data, i.e., the raw data before forming a SAR image
is used. By taking the phase history data at range
cell 393 and suppressing the ground clutter returns,
the micro-Doppler signature of the vibrating corner
reflector can be seen in Fig. 14(c). It shows that the
vibration rate of the corner reflector is 2.3 Hz, which
is close to the ground truth of 2.0 Hz. In [21], the
same data used in [20] is processed to calculate the
amplitude of the vibration that turns out to be 1.5 mm.
The micro-Doppler signature of the phase history data
at range cell 400 shown in Fig. 15(d) indicates that
the corner reflector at range cell 400 is vibrating at
multiple vibration rates, which is coincident to the
ground truth.
B. Micro-Doppler Signature of Human Walking Gait
Human walking is a complex motion that
comprises different movements of individual body
parts. The bulk motion of the body is basically a
translation with slightly body rocking and head
movement. The arms or legs swinging back and
forth are strong micro-motions described by a partial
rotation around an axis. Human experience shows
that it is possible to recognize a friend by gait [22].
However, gait recognition by machine is a challenge.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 1
JANUARY 2006
Fig. 15. Micro-Doppler signature of walking person with swinging arms.
Various computer vision and ultrasound techniques
have been developed to measure gait parameters
[23—28]. Radar gait analysis has recently been under
development [29—32]. Radar has the advantage of
detecting and identifying humans at distances in all
weather conditions at day or night.
The micro-Doppler effect in radar returns from a
walking person was studied in the late 1990s using
data collected by a roof-top X-band high-resolution
radar and reported in [29]. References [30] and
[31] also studied the potential of using radar for
gait analysis, where a fully coherent X-band CW
radar was used to record signatures of human gait
that contain rich information on the various parts of
the moving body for human gait recognition. They
simulated and analyzed gait cycle formed by the
movements of various individual body parts. The
gait signature was analyzed by the STFT. From the
time-frequency signature of the human gait, each
forward leg-swing can be seen by large peaks, and
the left and right leg-swing completes a gait cycle.
The body motion that is the stronger component
underneath the leg-swings tends to have a slightly
saw-tooth shape because the body speeds up and
slows down during the swing. From the signature of
a single gait cycle, the lower leg “eye hook” feature
above the main body’s Doppler frequency shift can be
seen.
Recently, [32] reported the current progress on
micro-Doppler analysis of human gait using radar.
A human walking model and the detailed movement
information on individual parts of a human body were
proposed.
The measured radar data used in this paper was
collected by a roof-top X-band radar (data provided
by Norden Systems, Northrop Grumman). A man
was walking towards the building at a speed of
about 1.8 m/s as illustrated in Fig. 15(a). The range
profile has 64 range cells and 1024 pulses were
collected at a PRF of 800 Hz. Fig. 15(b) shows a
radar range-Doppler image of the walking man using
64 range-cells and 128 pulses, where the hot spot in
the image indicates the body of the walking man. One
also notices a smeared line running across the Doppler
direction around the body of the walking man at range
cell 12.
By applying the STFT to the complex phase
history data at range cell 12, the body Doppler
shift and the micro-Doppler modulation of the
swinging arms can be clearly detected in the joint
time-frequency signature. The Doppler shift of
one arm is higher and the other is lower than the
CHEN ET AL.: MICRO-DOPPLER EFFECT IN RADAR: PHENOMENON, MODEL, AND SIMULATION STUDY
17
body Doppler frequency shift. Superposition of the
time-frequency signatures over all range cells, which
corresponds to different walking steps, gives a full
time-frequency signature of the walking man as shown
in Fig. 15(c). We can see that the body’s Doppler shift
is almost constant with a slightly saw-tooth shape
but the arm’s micro-Doppler shift is a time-varying
periodic curve. From the available time information,
the swinging rate of the arm can be estimated to about
1.2 cycle/s.
3D rotation matrix), which are defined by:
SO(3) = fR 2 <3£3 j R T R = I, det(R) = +1g: (59)
Computing the derivative of the constraint R(t)R T (t)
= I with respect to time t, we obtain
T
_
R(t)R
(t) + R(t)R_ T (t) = 0
T
_
!ˆ = R(t)R
(t):
This paper introduced the micro-Doppler
phenomenon in radar, derived mathematical formulas
solving micro-Doppler modulations induced by
vibration, rotation, tumbling, and coning motions,
and applied point scatterer target’s model to
micro-Doppler simulation studies. By comparing
the theoretical results with simulation results, we
confirmed the effectiveness of our theoretical results.
We also described applications of micro-Doppler
analysis in radar and demonstrated examples of the
micro-Doppler effect using real radar data. From the
extracted micro-Doppler signatures, information about
target’s micro-motion dynamics, such as vibration rate
of the cornel reflector and swinging rate of arms, can
be obtained.
The usage of time-frequency analysis for
extracting micro-Doppler signatures in clutter and
noise environment, and applications of micro-Doppler
analysis in radar to biometric identification, engine
diagnosis, and detection of propeller modulations
using an experimental X-band micro-Doppler radar
will be reported in a separate paper in the near future.
APPENDIX
For any vector ~u = [ux , uy , uz ]T and ~r, define a skew
symmetric matrix
3
2
0
¡uz uy
7
6
(57)
û = 4 uz
0
¡ux 5 :
¡uy
ux
0
The cross product of two vectors can be computed
through matrix computation, that is,
2
3
uy rz ¡ uz ry
6
7
~p = ~u £~r = 4 uz rx ¡ ux rz 5
2
¡uy
ux ry ¡ uy rx
32 3
¡uz uy
rx
76 7
0
¡ux 5 4 ry 5 = û~r:
ux
0
(58)
rz
This definition is very useful in analysis of SO(3)
groups (special orthogonal matrix groups, also called
18
T
T
_
_
R(t)R
(t) = ¡[R(t)R
(t)]T :
T
_
The result reflects the fact that the matrix R(t)R
(t) 2
3£3
<
is a skew symmetric matrix. Then there must
exist a vector ~! 2 <3 such that:
VI. SUMMARY
0
6
= 4 uz
)
(60)
Multiplying both sides by R(t) to the right yields:
_ = !R(t):
ˆ
R(t)
(61)
Assume that the vector ~! 2 <3 is constant. According
to the linear ordinary differential equation (ODE), we
can obtain:
ˆ
R(t) = expf!tgR(0)
(62)
ˆ
where expf!tgis
the matrix exponential:
ˆ = I + !t
ˆ +
expf!tg
ˆ 2
ˆ n
(!t)
(!t)
+ ¢¢¢ +
+ ¢ ¢ ¢ : (63)
2!
n!
Assuming R(0) = I for the initial condition we must
have
ˆ
R(t) = expf!tg:
(64)
ˆ is indeed a
One can conform that the matrix expf!tg
rotation matrix. Since
ˆ = exp(!ˆ T t) = [exp(¡!t)]
ˆ T
[exp(!t)]¡1 = exp(¡!t)
(65)
T
ˆ
ˆ = I, from which one can
then [exp(!t)]
(exp(!t))
ˆ = §1. Furthermore,
obtain det[exp(!t)]
·
µ ¶
µ ¶¸
ˆ
ˆ
!t
!t
ˆ = det exp
det[exp(!t)]
¢ exp
2
2
½ ·
µ ¶¸¾2
ˆ
!t
¸0
= det exp
2
ˆ = +1. Therefore,
which shows that det[exp(!t)]
ˆ is the 3D rotation matrix. Let
matrix R(t) = expf!tg
− = k~
! k. A physical interpretation of the equation
ˆ is simply a rotation around the axis
R(t) = expf!tg
!
~ 2 <3 by −t rad. If the rotation axis and the scalar
angular velocity are given by vector !
~ 2 <3 , the
ˆ
rotation matrix can be computed as R(t) = expf!tg
at time t.
Rodrigues’s formula is one efficient way to
ˆ
compute the rotation matrix R(t) = expf!tg.
Given
!
~ 0 2 <3 with k~!0 k = 1 and ~! = −~!0 , it is simple to
verify that the power of !ˆ 0 can be reduced by the
following formula:
!ˆ 03 = ¡!ˆ 0 :
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 1
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JANUARY 2006
Then the exponential series
ˆ = I + !t
ˆ +
exp(!t)
ˆ 2
ˆ n
(!t)
(!t)
+ ¢¢¢ +
+ ¢¢¢
2!
n!
[10]
(67)
[11]
can be simplified as
µ
¶
(−t)3 (−t)5
ˆ = I + −t ¡
exp(!t)
+
¡ ¢ ¢ ¢ !ˆ 0
3!
5!
¶
µ
2
4
(−t)
(−t)
(−t)6
¡
+
¡ ¢ ¢ ¢ !ˆ 02
+
2!
4!
6!
= I + !ˆ 0 sin −t + !ˆ 02 (1 ¡ cos −t):
[12]
(68)
[13]
Therefore
ˆ = I + !ˆ 0 sin −t + !ˆ 02 (1 ¡ cos −t):
R(t) = exp(!t)
(69)
[14]
ACKNOWLEDGMENTS
The authors thank Norden Systems, Northrop
Grumman for providing X-band radar data for
micro-Doppler analysis.
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Victor C. Chen received Ph.D. degree in electrical engineering from Case
Western Reserve University, Cleveland, OH in 1989.
Since 1990 he has been with Radar Division, U.S. Naval Research Laboratory,
Washington D.C., working on radar imaging, time-frequency applications to
radar, ground moving target indication, and micro-Doppler analysis. He served
as a technical representative he collaborating with several research groups
in universities and monitored a number of Navy SBIR projects. As a project
manager, he collaborates with international participants for an ONR NICOP
S&T project. His current research interests include computational synthetic
aperture radar imaging algorithms, time-frequency analysis for radar imaging
and signal analysis, SAR GMTI, radar micro-Doppler analysis, and kernel
machine/independent component analysis (ICA) for noncooperative target
identification.
Dr. Chen serves as a program committee member and session chair for
IEEE and SPIE conferences and served as a guest editor for IEE Proceedings
on Radar, Sonar and Navigation in 2003, a guest editor for the European Applied
Signal Processing Journal in 2005, and associate editor for IEEE Transactions
on Aerospace and Electronic Systems since 2004. Dr. Chen received NRL
Review Award in 1998, NRL Alan Berman Research Publication award in
2000 and 2004, and NRL Technical Transfer Award in 2002. He has more
than 100 publications in books, journals, and proceedings including a book:
Time-Frequency Transforms for Radar Imaging and Signal Analysis (V. C. Chen
and Hao Ling), Artech House, Boston, MA, January 2002.
Fayin Li received the B.S. degree in automatic control from Huazhong University
of Science and Technology, China, in 1996, and the M.S. degree in computer
science from the Institute of Automation, Chinese Academy of Sciences, China,
in 1999.
He is currently completing the Ph.D. degree in computer science at George
Mason University, Fairfax, VA. His research interests include computer vision,
machine learning, objects/location recognition, human-computer interaction,
robotics, image processing, pattern recognition, and data mining. He is also
interested in time-frequency analysis and its application.
20
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 1
JANUARY 2006
Shen-Shyang Ho received the B.Sc. degree in mathematics and computational
science from the National University of Singapore, in 1999, the M.Sc. degree in
computer science from George Mason University, 2003, and is currently pursuing
Ph.D. degree in computer science from George Mason University.
His current interests include online learning from data streams, adaptive
learning and pattern recognition.
Harry Wechsler (F’92) received the Ph.D. in computer science from the
University of California, Irvine, in 1975.
Currently, he is a professor of computer science and director for the Center of
Distributed and Intelligent Computation at George Mason University. His research
in the field of intelligent systems focuses on computer vision, machine learning,
and pattern recognition, with applications for active learning, biometrics/face
recognition, clustering and concept drift/change, data mining, augmented
cognition and intelligent HCI, performance evaluation, transduction, and video
tracking and surveillance.
Dr. Wechsler has published more than 200 scientific papers and the book
Computational Vision (Academic Press, 1990). As a leading researcher in
face recognition, he organized the NATO Advanced Study Institute on “Face
Recognition: From Theory to Applications” (1997), whose seminal proceedings
were published by Springer (1998). He also directed the development and
collection of FERET, which has become the standard facial data base for
benchmark studies and experimentation. He was elected an IEEE fellow in
1992 for “contributions to spatial/spectral image representations and neural
networks and their theoretical integration and application to human and machine
perception”; and an IAPR (International Association of Pattern Recognition)
fellow in 1998 for contributions to “computer vision and pattern recognition.” He
was granted (with his former doctoral students) two patents by USPO in 2004 on
fractal image compression using quad-Q-learning and feature based classification
(for face recognition).
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